Effects of Merging and Diverging on Freeway Traffic Oscillations
by
Soyoung Ahn (corresponding author)
Department of Civil and Environmental Engineering
Arizona State University
P.O. Box 875306
Tempe, AZ 85287-5306
Phone: (480) 965-1052
Fax: (480) 965-0557
[email protected]
Michael J. Cassidy
Department of Civil and Environmental Engineering and
Institute of Transportation Studies
University of California, Berkeley
416C McLaughlin Hall, Berkeley, CA 94720
Phone: (510) 642-7702
Fax: (510) 642-1246
[email protected]
and
Jorge Laval
Assistant Professor
Department of Civil and Environmental Engineering
Georgia Institute of Technology
790 Atlantic Drive N.W.
Atlanta, GA 30332-0355
Phone: (404) 894-2360
[email protected]
ABSTRACT
This study proposes a theory for explaining the growth of oscillations as they propagate through
congested merges and diverges. The idea is that merging or diverging flows change in response
to freeway oscillations, and these changes in flow have an effect of dampening (merging) or
amplifying (diverging) oscillations. In this theory, a reduction or an increase in amplitude is
quantified based on a single parameter (a fraction of entering flow for the merging effect and a
fraction of exiting flow for the diverging effect). The effect of merging has been verified at
multiple freeway merges where both the freeway and the on-ramps were saturated by abiding
queues for extended time. The findings show that oscillations diminished in amplitude by the
amount close to what the proposed theory predicts.
Effects of Merging and Diverging on Freeway Traffic Oscillations
1. INTRODUCTION
Traffic oscillations are stop-and-go driving motions that arise in queued traffic. These
disturbances propagate upstream through the queue, against the flow of traffic, and typically
exhibit changes in amplitude (they grow or shrink) as they do so (Mauch and Cassidy, 2002). In
an earlier empirical study (Ahn and Cassidy, 2007), freeway traffic oscillations were shown to
form and grow due to vehicle lane-change maneuvers. The present paper shows how merging
and diverging maneuvers near ramps further impact oscillation growths.
We present a theory whereby merging traffic from a fully queued, uncontrolled (e.g. un-metered)
on-ramp reduces oscillation amplitudes and thus counters the effects of the lane-changing that
occurs away from ramps. We further theorize that these resulting reductions in amplitude are
proportional to the fraction of entering traffic. The merging effect can thus be predicted in a
simple way given the ratio that defines how drivers from the two conflicting stream (the freeway
approach and the on-ramp) take turns when merging (Daganzo, 1994; Cassidy and Ahn, 2006).
An analogous theory is offered for describing the effects of diverging traffic near off-ramps.
We furnish empirical evidence in support of the proposed theory. Traffic data were taken from
multiple merge locations where the on-ramp and freeway at each were congested. The data show
that at each location, the amplitude of freeway oscillations diminished as predicted by our
proposed theory. This work is notable in that oscillations have long been theorized as the results
of instabilities in car-following behavior, independent of merging and diverging flows.
The following section summarizes previous work related to oscillations. In section 3, our theory
pertaining to merging and diverging effects is presented in detail. General features of
oscillations and the empirical validation of the proposed theory are discussed in section 4,
including the descriptions of sites and data. Concluding remarks are offered in section 5.
2. BACKGROUND
Traffic oscillations have long been described as part of car-following theories that are formulated
with the interaction between a lead vehicle and its immediate follower. Details aside, numerous
models have been developed (e.g., Chandler et al., 1958; Kometani and Sasaki, 1959; Michaels,
1963), calibrated (e.g., May and Keller, 1967; Ceder and May, 1976; Aron, 1988), and modified
(e.g. Edie, 1960; Gipps, 1981) since the 1950’s. In these models, oscillatory flows arise when
instabilities in vehicular interaction occur within a single traffic stream (Herman and Montroll,
1959). Some empirical evidence has been found in support of the car-following theories, such
that oscillations arose and grew without vehicle lane-changing (Edie and Baverez, 1958;
Smilowitz et al., 1999; Sugiyama et al., 2003).
Mauch and Cassidy (2002), however, found empirical evidence against what car-following
models would predict and suggested that lane-changing maneuvers are a key player in the
development of oscillations. In particular, the data obtained from inductive loop detectors on a
multi-lane freeway displayed oscillations with periods of several minutes rather than several
seconds as early car-following models predict. Moreover, oscillations propagated against
congested traffic flow without losing their structure near freeway interchanges (also observed in
Kerner and Rehborn, 1996), and their amplitude grew in general along their paths (also observed
in Kerner, 1998). More surprisingly, oscillations tended to form and grow near freeway ramps,
Ahn, Cassidy, and Laval
1
Effects of Merging and Diverging on Freeway Traffic Oscillations
where many systematic lane-changing maneuvers abounded (and hence, car-following theories
do not typically hold). These studies, however, did not offer explanations on how the effect of
lane-changing maneuvers compares to the car-following effect, and how merging and diverging
play distinctive roles in oscillation growths.
The causal link among lane-changing maneuvers, car-following effects, and the evolution of
oscillations was elaborated in Ahn and Cassidy (2007). Their macro-level analysis based on
loop detector data confirmed many of the earlier findings, such that oscillations exhibited periods
of several minutes and grew in amplitude as they propagated upstream. More notably, the study
entailed micro-level analysis by extracting vehicle trajectories from video. Using these
trajectories, several instances of oscillation formations and growths were systematically detected.
These detected instances revealed that lane-changing maneuvers were triggering events for both
formation and growth and that car-following effects followed once an oscillation was triggered.
A similar finding has been presented in Laval (2005), who used the simulation model in Laval
and Daganzo (2006) to show that lane-changing is the source of oscillations.
Ahn and Cassidy (2007)’s macro-level study revealed an interesting feature pertaining to
oscillation growth. Unlike the earlier studies, the amplitude of oscillations diminished near busy
on-ramps despite large systematic lane-changing maneuvers induced by merging. The reduction
in amplitude was especially noticeable near an on-ramp that typically exhibits long queues
during the rush. This unique finding seemed attributable to the freeway site selected for analysis,
in that the on-ramps in the study site are not controlled by ramp meters. (The freeway site
analyzed in Mauch and Cassidy (2002) contains a number of controlled on-ramps. Although the
metering rates are designed to change in response to freeway conditions, the rates are nearly
constant during the height of rush-hour.) Motivated by this finding, the present paper proposes a
theory pertaining to the effects of merging and diverging on oscillation growths in ways distinct
from that of lane-changing. Empirical evidence found in support of the proposed theory is
provided as well.
3. PROPOSED THEORY
This section provides a theory that describes how merging counters the effects of lane-changing
and modifty the amplitude of oscillations. The analogous theory for diverging effects is
presented in this section as well. In the theory offered here, ramp flows at congested merges and
diverges change in response to freeway oscillations. The changes in merging or diverging rates,
in turn, affect the oscillations’ amplitudes.
Merging Effect
The illustration in Fig. 1(a) shows a hypothetical merge with its freeway and on-ramp engulfed
in queues. We assume that the freeway’s traffic conditions are described by a fundamental
diagram, like the piece-wise linear relation shown in Fig. 1(b).
It is further assumed that freeway conditions just downstream of the merge (at the location
labeled xD) are given by the point labeled D on the fundamental diagram. The queued flow there,
qD, is the sum of qU, the freeway inflow at upstream location xU, and qon, inflow from the onramp. These inflows, annotated in Fig. 1(b), are determined by the ratio that freeway and onramp drivers take turns at the congested merge (Daganzo, 1994).
Ahn, Cassidy, and Laval
2
Effects of Merging and Diverging on Freeway Traffic Oscillations
This above-cited merge ratio is defined as  = qon / qU. It is assumed constant, independent of
the flow at xD, as per empirical findings (Cassidy and Ahn, 2006). Thus, inflows qU and qon at
1

the merging point are
qD and
qD, respectively.
1
1
Traffic states at the merge are perturbed by incoming oscillations that arrive there. Fig. 1(b)
depicts the case of a flow perturbation of positive magnitude D. Queued merge outflow rises
from qD to qD' and the traffic state at xD moves from D to D'.
The figure further shows that queued inflows to the merge increase to qU' and qon'. They do so in
such ways that  remains constant; qon' / qU' = qon / qU. Since
qD' = qD + D ,
and since the perturbation in freeway inflow at xU, U, is

U = qU' – qU =
1
1
qD' –
qD ,
1
1
1
|D |. Since  is non-negative, | U | < | D | for any value
1
Furthermore, the perturbation (and thus the oscillation’s amplitude) diminishes by
its absolute value, | U | equals
of .
100
percent as it propagates through the merge. Physically, the oscillation is partially absorbed
1 
by the flow on the ramp, such that the flow change at xU is less than the change at xD.
Diverging Effect
An analogous logic can be applied to a diverge engulfed in a freeway queue. With this geometry,
oscillations grow in amplitude as they propagate upstream.
A hypothetical site is shown in Fig. 2(a). Since part of the outflow from the diverge goes to the
ramps, queued flow at xU exceeds that measured at xD. Thus, state D lies below state U on the
fundamental diagram, as in Fig. 2(b). The vertical difference, qoff, is the exiting flow through the
off-ramp.
When the perturbation downstream, D, is positive, the new off-ramp flow qoff´ is larger than the
initial flow, which magnifies the flow perturbation to the freeway upstream (D<U) as shown in
Figure 2(b). Assuming a fixed fraction, , of exiting flow, U is derived as
U = qU' – qU =
1
1
1
qD' –
qD =
D
1 
1 
1 
Ahn, Cassidy, and Laval
3
Effects of Merging and Diverging on Freeway Traffic Oscillations
Since 0 < < 1, | U | > | D | for any value of , and the percent increase in amplitude is
100
.
1 

4. EMPIRICAL FINDINGS
This section presents the data that reveal general features of oscillations and empirical evidence
found in support of the proposed theory.
General Oscillatory Features
In order to see the general attributes of oscillatory traffic, we analyzed the data from a 5-km-long
stretch of eastbound Interstate 80 near San Francisco, California (see Fig. 3). During afternoon
rush periods, the site has a major bottleneck at its downstream end. The resulting queue
(represented in the figure with shading) fills the entire freeway stretch. Freeway flows within
this queue varied with location (from about 7,000 to 8,000 vph) due to inflows and outflows
from the ramps.
Flows, as well as occupancies and time-mean average vehicle speeds, were measured over 10second sampling intervals at the eight detector stations labeled in the figure. In addition, a video
surveillance system provided unobstructed views of traffic over a long portion of the site.
General attributes of oscillatory flow were unveiled by processing the loop detector data in ways
described below. These data were measured during 1-hr periods of full freeway congestion on
each of 5 days in June and July 2003.
Fig. 4(a) presents data from a typical study period (on June 25, 2003). Shown in this figure are
specially transformed curves of cumulative vehicle count vs. time. There is one curve for each
detector station and at each station, the curve was constructed from the queued vehicle counts
across all regular-use freeway lanes. What these curves actually display are N t   N 5 t  , the
vertical deviations between the cumulative count at time t, N(t), and the 5-minute moving
average of the counts spanning that t, N 5 t  ; i.e., each deviation was computed as
N t   N t  2.5 min  N t  2.5 min / 2 .
The curves are drawn as piece-wise linear interpolations to the 10-second detector counts, such
that the curves’ slopes are differences from longer-run average flows. Thus, the wiggles clearly
evident in the curves are oscillations. Positive slopes reveal periods of vehicle acceleration.
Negative slopes mark deceleration periods. Fig. 4(a) shows that each such period was typically
several minutes in duration.
Of further note, dotted lines have been added to Fig. 4(a) to trace (approximately) the motion of
oscillatory waves (wiggles) as they propagated against the flow of queued traffic. That these
lines are straight and roughly parallel (despite changes in flow over space) indicates that wave
speed was approximately independent of flow1.
1
The average speeds at which oscillatory waves propagated over the 5-km-long stretch of queued freeway were
nearly fixed; they ranged from 18 to 21 km/hr.
Ahn, Cassidy, and Laval
4
Effects of Merging and Diverging on Freeway Traffic Oscillations
The above findings are consistent with those from an earlier study (Mauch and Cassidy, 2002).
And like the previous study, we have presently observed that oscillations often grow in
amplitude as they propagate through queued traffic.
Some of this detail is unveiled in Fig. 4(b). To construct the figure, the Root Mean Squared
Error (RMSE) of the 10-sec flow deviations were calculated for each day and at each detector
station as
 360

2
 N t   N 5 t  / 360
 t 0

1/ 2
The figure’s lightly shaded circles are the averages of these RMSEs taken over the five rush
periods from which data were collected.
Visual inspection of Fig. 4(b) shows that for the downstream freeway segments between
detectors 1 – 3, the oscillations were relatively small in amplitude and display no real growth
over space. On these freeway segments so near the downstream bottleneck, the oscillations had
formed only recently and had yet to fully develop. Large growth from detectors 3 to 5 is
attributable to vehicle lane-changing maneuvers, as elaborated in Ahn and Cassidy (2007).
Fig. 4(b) also shows that oscillations diminished in amplitude further upstream, as they
propagated from detectors 5 to 6 and, to a lesser extent, while propagating between detectors 7
and 8. We next provide evidence that the (negative) growth pattern on these segments can be
attributed to inflows from the nearby on-ramps.
Empirical Validation
The theory was separately validated for two fully congested merges on eastbound Interstate 80
(Fig. 3). These were the merges formed by the on-ramps from Powell St. and from Ashby Ave.
Fig. 5(a) is a sketch of the Powell St. merge. Vehicle counts were taken (from videos) at the
three locations labeled x0 – x2 in the figure. These were taken during a 50-minute period on
August 19, 2002 when both the freeway and on-ramp were queued.
Fig. 5(b) presents curves of N t   N 5 t  ; one is from all regular-use freeway travel lanes at x1;
the other from the on-ramp (only) at x0. These deviation curves show that oscillations occurring
on the ramp were much smaller in amplitude than those downstream at x1. This was to be
expected since the ramp has only a single lane and its vehicles share available merge capacity
with freeway traffic.
The curves nevertheless show correlation with a time lag. The latter is the trip time of oscillatory
waves from x1 to x0; this time corresponded to a wave speed of 23 km/hr, a rate comparable to
those reported in empirical studies cited earlier. This cross correlation was estimated to be 0.78,
indicating that on-ramp inflows changed in response to freeway oscillations immediately
downstream.
Ahn, Cassidy, and Laval
5
Effects of Merging and Diverging on Freeway Traffic Oscillations
Fig. 5(c) shows the RMSEs computed for the flow deviations at each location, x0 – x2. As
oscillations propagated from x2 to x1, amplitudes increased, as is evident in the RMSEs that grew
from 19.6 to 20.1 vehs (a trend of 3.7 vehs/km). This trend reflects the effect of vehicle lanechanging maneuvers, since segment x1 – x2 lies downstream of the merge. The trend was
extrapolated to estimate what the RMSE would have been for flow deviations at x0, had merging
not occurred just downstream.
The difference between the projected RMSE (20.5 vehs, as shown in Fig. 5(c)) and the measured
value (17.7 vehs) was taken as the reduction due to merging. This difference, 2.8 vehs, is a
reduction of 14.3 percent. Notably, the amplitude reduction predicted by the theory was 15.1
percent, a value close to the estimated reduction. (The prediction was made by estimating the
merge ratio from data in the manner described in Cassidy and Ahn, 2006. For this location, the
estimated merge ratio was 0.178.)
The theory also furnished reliable predictions for the merge formed by the Ashby Ave. on-ramp.
By referring back to Fig. 4(b), the reader will note that the projected RMSE for deviations at
detector 6 (near the merge) is 26.1, such that the RMSE reduction due to merging was taken to
be 28.0 percent. The theory predicted an amplitude reduction of 27.1 percent based on the
estimated merge ratio of 0.371.
5. CONCLUSIONS
This paper described a theory pertaining to the effect of merging and diverging traffic flow on
the spatial growth of freeway oscillations, whereby merging dampens oscillations and diverging
amplifies them. Data extracted near fully congested merges verified that the amplitude of
oscillations diminished near these locations, and the corresponding percent reduction was
consistent with what the proposed theory predicts.
The findings indicate that this proposed theory, in conjunction with the theory of merge by
Daganzo (1994), can be used to quantify the changes in oscillations’ amplitudes near congested
merges in a simple manner (by estimating a merge ratio while a merge is fully-congested).
Moreover, this theory can be extended to accommodate different merging schemes, resulting
from a ramp-metering system that controls entering traffic, for example.
For the effect of diverging flows, the freeway site used in this study did not provide suitable
diverges for analysis. Oscillations near one of the (two) off-ramps were not quite developed (i.e.
they were asynchronous across lanes) because of the ramp’s close proximity to the active
bottleneck. The other off-ramp is located just upstream of freeway lane reduction, and hence the
effect of diverging flow could not be analyzed independently. Therefore, a verification of the
effect of diverging flow is left for future study. Nevertheless, this present study provides an
important insight as to how oscillations behave near inhomogeneous sections by merges and
diverges, which are common features of freeway systems.
Ahn, Cassidy, and Laval
6
Effects of Merging and Diverging on Freeway Traffic Oscillations
REFERENCES:
Aron, M. (1988). Car Following in an Urban Network: Simulation and Experiments. In Proc.
of Seminar D, 16th PTRC Meeting, pp. 27-39.
Ahn, S. and Cassidy, M. J. (2007). Freeway Traffic Oscillations and Vehicle Lane-Change
Maneuvers. 17th Int. Symp on Traffic and Transportation Theory, Accepted.
Ahn, S. (2005). Formation and Spatial Evolution of Traffic Oscillations. Doctoral
Dissertation. University of California, Dept. of Civil and Environmental Engineering. Berkeley,
U.S.A.
Cassidy, M. J. and Ahn, S. (2006). Driver Turn-Taking Behavior in Congested Freeway
Merges. Transportation Research Record, 1934, pp. 140-147.
Ceder, A. and May A. D. (1976). Further Evaluation of Single and Two Regime Traffic Flow
Models. Transportation Research Record, 567, pp. 1-30.
Chandler, R. E., Herman, R. and Montroll, E. W. (1958). Traffic Dynamics: Studies in Car
Following. Operations Research 6, pp. 165-184.
Daganzo, C. F. (1994). The cell transmission model: A dynamic representation of highway
traffic consistent with the hydrodynamic theory, Transportation Research 28B, pp. 269-287.
Edie, L. C. (1960). Car Following and Steady State Theory for Non-Congested Traffic.
Operations Research 9, 66-76.
Edie, L. C. and Baverez, E. (1958). Generation and Propagation of Stop-Start Traffic Waves.
Proc. Of the 3rd International Symposium on Theory of Traffic Flow (L.C. Edie, Ed.), American
Elsevier Publishing Co., New York, 26-37
Gipps, P. G. (1981). A Behavioral Car-following Model for Computer Simulation.
Transportation Research 15B, pp. 105-111.
Herman, R., Montroll, E. W., Potts, R. B., Rothery R. W. (1959). Traffic Dynamics: Analysis
of Stability in Car Following. Operations Research 7, pp. 86-106.
Kerner, B. S. (1998). Experimental Features of Self-Organization in Traffic Flow, Physical
Rev. Lett. 81(17), pp. 3797-3800
Kerner, B. S. and Rehborn, H. (1996). Experimental Features and Characteristics of Traffic
Jams, Physical Review E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary
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Effects of Merging and Diverging on Freeway Traffic Oscillations
Kometani, E. and Sasaki, T. (1959). Dynamic Behavior of Traffic with a Nonlinear SpacingSpeed Relationship. Proc. of the Symposium on Theory of Traffic Flow. Elsevier, Amsterdam,
pp. 105-119, Herman, R. (Ed.),
Laval, J. A. (2005). Linking synchronized flow and kinematic wave theory. In R. Kuhne, T.
Poschel, A. Schadschneider, M. Schreckenberg, and D. Wolf, editors, Forthcoming in Traffic
and Granular Flow'05. Springer, 2005.
Laval, J. A. and Daganzo, C. F. (2006). Lane-Changing in Traffic Streams. Transportation
Research B, 40(3), pp. 251-264.
Mauch, M. and Cassidy, M. J. (2002). Freeway Traffic Oscillations: Observations and
Predictions. Procs. 15th Int. Symp on Traffic and Transportation Theory, M. P. Taylor (ed.)
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Michaels, R. M. (1963). Perceptual Factors in Car Following. Proc. of the 2nd Int. Symp on the
Theory of Road Traffic Flow (pp. 44-59). Paris: OECD.
Smilowitz K., Daganzo, C., Cassidy M., and Bertini R. (1999). Some observations of highway
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Sugiyama, Y., Nakayama, A., Fukui, M., Hasebe, K., Kikuchi, M., Nishinari, K., Tadaki, S. and
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Traffic and granular flow '03, pg. 45-59
Ahn, Cassidy, and Laval
8
Effects of Merging and Diverging on Freeway Traffic Oscillations
Direction of
oscillatory
wave
flow
qmax
XD
qD'
 U < D
D
D'
qD
qU'
qU
XU
qon′ > qon
D
U
U'
qon
U
density
Travel
direction
(a)
kjam
(b)
FIGURE 1
(a) Hypothetical Freeway Merge
(b) Theory of Merging Effect
Ahn, Cassidy, and Laval
9
Effects of Merging and Diverging on Freeway Traffic Oscillations
Direction of
oscillatory
wave
flow
qmax
XD
qU'
qU
qD'
qD
XU
 U > D
U
U'
qoff′ > qoff
U
D
D'
qoff
D
density
kjam
Travel
direction
(a)
(b)
FIGURE 2
(a) Hypothetical Freeway Diverge
(b) Theory of Diverging Effect
Ahn, Cassidy, and Laval
10
Effects of Merging and Diverging on Freeway Traffic Oscillations
I-580W
I-80E
Bottleneck
Buchanan St.
High-occupancy
vehicle (HOV) Lane
(3pm -7pm)
Gilman St.
460m
1
460m
2
University Ave.
lane 1
lane 2
lane 3
lane 4
lane 5
550m
3
Travel
Direction
550m
4
6
Ashby Ave.
640m
7
340m 520m
5
8
Powell St.
FIGURE 3 Eastbound Interstate 80 near San Francisco
Ahn, Cassidy, and Laval
11
Ahn, Cassidy, and Laval
Loop
detector
stations
8
7
6
5
4
3
2
1
Travel
Direction
Powell
1500
vph
Ashby
600
vph
900
vph
Direction of
oscillatory wave
Deviation at each detector station, N(t) – N5(t)
FIGURE 4
13
15
(b) Root mean squared error measured at loop detector stations
(a) Deviation curves at loop detector stations on I-80E
(a)
Time
17:00 17:05 17:10 17:15 17:20 17:25 17:30 17:35 17:40 17:45 17:50 17:55 18:00
50
11
17
21
(b)
(20.1)
(21.4)
19
RMSE (vehs)
25
27
28.0% (26.1)
projection
23
Effects of Merging and Diverging on Freeway Traffic Oscillations
12
Ahn, Cassidy, and Laval
x0
120m
x1
140 m
x2
Direction of
oscillatory wave
(b)
(a)
20
18
(17.7)
17
FIGURE 5
(a) Eastbound Interstate 80 near Powell St. on-ramp merge
(b) Deviation curves at the measurement locations
(c) Root mean squared error computed at the measurement locations
Time
17:30 17:35 17:40 17:45 17:50 17:55 18:00 18:05 18:10 18:15 18:20
on-ramp
freeway lanes
Travel
Direction
Powell
St.
Deviations, N - N5
(c)
14.3%
3.7
vehs/km
19
21
(20.5)
(20.1)
(19.6)
20
RMSE (vehs)
Effects of Merging and Diverging on Freeway Traffic Oscillations
13